Derivatives - AP Calculus AB (2024)

Correct answer:

Explanation:

Differentiate,

Strategy

This one at first glance appears difficult even if we recognize that the chain rule is needed; we have a function within a function within a function within a function. To avoid making mistakes, it's best to start by defining variables to make the calculation easier to follow.

Let's start with the outermost function, we will writeas a function ofby setting,

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Similarly, define to writeas a function of

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Writeas a function of

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Finally, define the inner-most function,,as the function of

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Sincewe will just substitute that in and move to the front.

That was easy enough, now just write everything in terms ofby going back to the definitions ofand.

Correct answer:

Explanation:

To find the tangent line at the given point, we need to first take the derivative of the given function.

Power Rule:

Power rule says that we take the exponent of the “x” value and bring it to the front. Then we subtract one from the exponent.

Therefore,becomes

From there we plug in the "1" from the point to get our m value of the equation. When we plug in "1" to y' we get m=-1. Then from there, we will plug our point intonow that we have found m to find our b value. So,

Therefore, the tangent line is equal to

Correct answer:

Explanation:

To find the tangent line at the given point, we need to first take the derivative of the given function. The rule for functions with "e" in it says that the derivative ofHowever with this function there is also a 3 in the exponent so we will also use chain rule. Chain Rules states that we work from the outside to the inside. Meaning we will take the derivative of the outside of the equation and multiply it by the derivative of the inside of the equation.

To put this into equation it will look like

From there we plug in the "0" from the point to get our m value of the equation. When we plug in "0" to y' we get m=3. Then from there, we will plug our point intonow that we have found m to find our m value. So,

then plug this all back into the equation once more and we are left with

Correct answer:

Explanation:

To find the tangent line at the given point, we need to first take the derivative of the given function using Power Rule

Power rule says that we take the exponent of the “x” value and bring it to the front. Then we subtract one from the exponent.

From there we plug in the "2" from the point to get our m value of the equation. When we plug in "2" to y' we get m=8. Then from there, we will plug our point intonow that we have found m to find our bvalue. So,

Plug this back into

Correct answer:

Explanation:

Find the equation of the line tangent to the curve at the given point

The slope of the line tangent at the given pointwill be equal to the derivative of at that point. Compute the derivative and find the slope for our line:

Evaluate the secant term:

Therefore slope of the tangent line is simply:

So now we know the slope of the tangent line and can write the equation then solve for

In order to solve for we need one point on the line. Use the point where the tangent line meets the curve. Use the original function to find the "y" coordinate at this point:

We now have our point:

Use the point to find

Correct answer:

Explanation:

Find the slope of the line tangent to the curve of d(g) when g=6.

All we need here is the power rule. This states that to find the derivative of a polynomial, simply subtract 1 from each exponent and then multiply each term by their originalexponent.

Constant terms will drop out when we do this, and linear terms will become constants.

From here substitute in g=6.

Correct answer:

Explanation:

The tangent line to the graph ofat pointis the line with slopethat passes through that point. Findthe derivative:

Apply the sum rule:

The tangent line is therefore the line with slope 5 through.Apply the point-slope formula:

None of the other choices gives the correct response.

Correct answer:

None of the other choices gives the correct response.

Explanation:

The tangent line to the graph ofat pointis the line with slopethat passes through that point. Findthe derivative:

Apply the constant multiple and sum rules:

Setandand apply the chain rule.

Substituting back:

Evaluateusing substitution:

The tangent line is therefore the line with slope through.is a-intercept, so apply the slope-intercept formula to get the equation

.

This is not among the choices given.

Correct answer:

Explanation:

We first start by finding the slope of the line in question, which we do by taking the derivative ofand evaluate at.

,

We then use point slope form to get the equation of the line at the point

Correct answer:

Explanation:

The first step is to find the derivative of the function given, which is. Next, find the slope at (1,4) by plugging in x=1 and solving for , which is the slope. You should get. This means the slope of the new line is also -1 because at the point where a slope and a line are tangent they have the same slope. Use the equation to express your line. Y and x are variables and mis the slope, so the only thing you need to find is b. Plug in the point and slope into to get . Now you can express the general equation of the line as .